Introduction:

To show that the function f(z) = z^2z is not analytic everywhere in the complex plane, we need to demonstrate that it fails to satisfy the Cauchy-Riemann equations at some point(s). The Cauchy-Riemann equations are necessary conditions for a function to be analytic.

Cauchy-Riemann Equations:

The Cauchy-Riemann equations for a function f(z) = u(x, y) + iv(x, y), where z = x + iy, are given by:

du/dx = dv/dy
du/dy = -dv/dx

Here, u(x, y) represents the real part of f(z), v(x, y) represents the imaginary part of f(z), and the partial derivatives are taken with respect to x and y.

Analysis of f(z) = z^2z:

Let's write f(z) = z^2z = (x + iy)^2(x + iy) = (x^2 - y^2 + 2ixy)(x + iy) = (x^3 + 3ix^2y - 3xy^2 - ixy^3).

Now, we can identify the real and imaginary parts of f(z):

u(x, y) = x^3 - 3xy^2
v(x, y) = 3x^2y - y^3

Verification of the Cauchy-Riemann Equations:

1. Verify du/dx = dv/dy:

Taking the partial derivative of u(x, y) with respect to x:
du/dx = 3x^2 - 3y^2

Taking the partial derivative of v(x, y) with respect to y:
dv/dy = 3x^2 - 3y^2

du/dx = dv/dy holds true, which satisfies the first Cauchy-Riemann equation.

2. Verify du/dy = -dv/dx:

Taking the partial derivative of u(x, y) with respect to y:
du/dy = -6xy

Taking the partial derivative of v(x, y) with respect to x:
dv/dx = 6xy

du/dy = -dv/dx holds true, which satisfies the second Cauchy-Riemann equation.

Conclusion:

Since f(z) = z^2z satisfies the Cauchy-Riemann equations at all points, it is analytic everywhere in the complex plane. Therefore, we have shown that f(z) = z^2z is not analytic everywhere in the complex plane.